Theorem dmdi2 32338

Description: Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)

Assertion

Ref	Expression
dmdi2	⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵))

Proof of Theorem dmdi2

Step	Hyp	Ref	Expression
1		dmdi 32336	. 2 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)))
2		eqimss2 4068	. 2 ⊢ (((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) → (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵))
3	1, 2	syl 17	1 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ∩ cin 3975   ⊆ wss 3976   class class class wbr 5166  (class class class)co 7450   C_ℋ cch 30963   ∨_ℋ chj 30967   𝑀_ℋ^* cdmd 31001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-iota 6527  df-fv 6583  df-ov 7453  df-dmd 32315

This theorem is referenced by: (None)